Hybrid system for demand prediction

ABSTRACT

In demand prediction, a history of demand for a resource is modeled to generate a baseline model of the demand, and demand for the resource at a prediction time is predicted by evaluating a regression function of depth k operating on an input data set including at least the demand for the resource at the prediction time output by the baseline model and measured demand for the resource measured at k times prior to the prediction time. The resource may be off-street parking, and the input data set may further include weather data. The regression function may comprise a support vector regression (SVR) function that is trained on the history of demand for the resource. The baseline model suitably comprises a Fourier model of the history of demand for the resource.

BACKGROUND

The following relates to the demand prediction arts such as off-streetparking demand prediction, predictive modeling of time series data, andrelated arts.

Some illustrative examples of demand prediction problems includepredicting demand for off-street parking, predicting demand for acommercial product (e.g. gasoline, electrical power, a retailmerchandise product), and so forth. Typically, such demand followscertain discernible patterns. For example, demand for off-street parkingcan be expected to exhibit daily cycles (e.g., parking usage as measuredby parking ticket issuances may be expected to be highest in the morningas commuters arrive and lower during the day, with the lowest usage themidnight-early a.m. time interval) weekly cycles (e.g. high parkingusage during the workweek and lower usage on weekends, or vice versa forparking facilities that mostly service weekend clientele), and longerseasonal cycles (e.g., a parking facility that services a university maybe heavily used when the university is in session and lightly usedbetween sessions).

By recognizing such cycles, a parking lot operator can plan ahead interms of scheduling parking lot attendants for work, optimizing parkingprices based on predicted demand, and so forth. Additionally, theparking lot operator may take known special events into account, forexample a parking facility located near a football stadium may beexpected to see heavy usage just prior to a football game held at thestadium.

BRIEF DESCRIPTION

In some embodiments disclosed herein, a non-transitory storage mediumstores instructions readable and executable by an electronic dataprocessing device to perform a method comprising: estimating demand fora resource at a prediction time based on a harmonic analysis of ahistory of the demand; and generating a prediction of demand for theresource at the prediction time by evaluating a predictor functionoperating on an input data set including at least the estimated demandfor the resource at the prediction time estimated based on the harmonicanalysis and measured demand for the resource measured at one or moretimes prior to the prediction time. The resource may, for example, beoff-street parking, electrical power, gasoline, or a retail merchandiseproduct. In some embodiments the predictor function comprises a supportvector regression (SVR) function trained on the history of the demand.In some embodiments the method further comprises generating a Fouriermodel of the history of the demand, for example by computing Fouriercomponents for different periods including at least two of (1) one day,(2) one week, and (3) one year, or by computing a Fourier transform ofthe history of the demand, and the estimating comprises estimatingdemand for the resource at the prediction time by evaluating the Fouriermodel at the prediction time.

In some embodiments disclosed herein, an apparatus comprises anelectronic data processing device configured to receive occupancy datafrom an off-street parking resource, the electronic data processingdevice further configured to: estimate demand for the off-street parkingresource at a prediction time based on a model of historical occupancydata received from the off-street parking resource; and generate apredicted demand for the off-street parking resource at a predictiontime by evaluating a regression function operating on an input data setincluding at least the estimated demand for the off-street parkingresource at the prediction time estimated based on the model andoccupancy data received from the off-street parking resource at k timesprior to the prediction time. The input data set may further includeweather data measured at the k times prior to the prediction time. Theregression function may comprise a SVR function trained on thehistorical occupancy data. The electronic data processing device may befurther configured to generate said model as a Fourier model of thehistorical occupancy data, for example by operations including computinga Fourier component for a time interval of one day and computing aFourier component for a time interval of one week.

In some embodiments disclosed herein, a method comprises modeling ahistory of demand for a resource to generate a baseline model of thedemand, and predicting demand for the resource at a prediction time byevaluating a regression function of depth k operating on an input dataset including at least the demand for the resource at the predictiontime output by the baseline model of the demand and measured demand forthe resource measured at k times prior to the prediction time, where kis a positive integer greater than or equal to one. The modeling and thepredicting are suitably performed by an electronic data processingdevice. In some embodiments k is greater than or equal to two. In someembodiments the resource is off-street parking and the input data setfurther includes weather data measured at the one or more times prior tothe prediction time. In some embodiments the method further comprisestraining the regression function, such as an SVR function, on thehistory of demand for the resource, wherein the training is performed bythe electronic data processing device and utilizes the baseline model ofthe demand. In some embodiments the baseline model comprises a Fouriermodel, and the modeling comprises computing Fourier components of thehistory of demand for the resource.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1 and 2 diagrammatically show a system for performing regressionanalysis of countable data, including a diagrammatic representation ofthe learning phase of the regression analysis (FIG. 1) and adiagrammatic representation of the inference phase (FIG. 2).

FIGS. 3-13 illustrate simulation/experimental results as describedherein.

DETAILED DESCRIPTION

Demand for products or services such as off-street parking is expectedto follow certain cyclical patterns, possibly modified by specificdiscernible events such as sports events in the case of a stadiumparking lot. However, it is recognized herein that cycle-based demandprediction (even augmented by taking into account recognizable specialevents) may not provide sufficiently accurate demand prediction,especially in mid- to short-term time intervals. Considering off-streetparking demand as an illustrative example, the demand may be impacted byother factors such as weather conditions, less significant specialevents (e.g. in the case of a city parking lot, it may be difficult toknow about, and/or predict the effect on parking demand of, a smallconference held some distance from the parking lot), current gasolineprices, and so forth. Failure to account for such latent factors canadversely impact parking lot profitability and operational efficiencythrough non-optimal pricing and inefficient scheduling of parking lotattendants.

It is disclosed herein to improve prediction of demand for a resource ata prediction time by estimating the demand based on a harmonic analysisof a history of the demand, and then generating a prediction of demandfor the resource at the prediction time by evaluating a predictorfunction operating on an input data set including at least the demandfor the resource at the prediction time estimated based on the harmonicanalysis and measured demand for the resource measured at one or moretimes prior to the prediction time. In this way, the demand estimategenerated by harmonic analysis is adjusted for shorter term variationsusing the predictor function, while still leveraging the harmonicanalysis estimate which is expected to be typically relatively accurate.

The resource can be, by way of non-limiting illustrative example:off-street parking, electrical power, gasoline, a retail merchandiseproduct, or so forth. By way of illustration, a system for predictingdemand for an off-street parking resource is described. Off-streetparking facilities are commonly constructed in high activity areas ornear high activity locations, such as airports, shopping malls,recreation areas, downtowns or other commerce centers, and so forth. Useof parking facilities by drivers is affected by various internal andexternal factors, and identifying and incorporating this information isadvantageous for understanding decisions made by drivers. Another factorthat may impact use of a parking resource is the pricing. The price forparking is typically a function of the time duration of use (e.g., acertain price per hour) and may also be a function of the time-of-day(e.g., an “early bird special” for parkers who arrive early). The pricesthat a particular parking facility can charge are impacted by theproximity to points of interest, overall availabilty and demand forparking in the locale, and so forth.

In the disclosed demand prediction for an off-street parking resource,the goal is predict the parking demand based on a model incorporatinghistorical occupancy data and other available observations. The demandprediction can serve diverse useful purposes, such as optimizing pricingto maximize profit, maximizing availability of parking spaces, providinga principled basis for parking lot attendant work shift scheduling, andso forth. These applications may be interweaved: for example, adjustingthe pricing based on the occupancy predictions can cause modificationsin driver behavior leading to reduction in traffic congestion andincrease availability of the limited parking resources during peakhours. The disclosed demand prediction incorporates historical data andoptionally other aspects which may impact the demand, such as seasonaleffects, calendar information (e.g. day-of-week, time-of-day, etcetera), weather conditions, status of adjacent sub-systems (rail, busesor flights), and so forth.

The historical data of parking occupancy is suitably represented as atime series, and may be modeled using various approaches such as machinelearning techniques (e.g. support vector regression, SVR),auto-regressive models like auto-regressive integrated moving average(ARIMA), spectral methods like harmonic decomposition, neural networks,regression trees, k-nearest neighbors, Gaussian processes, and so forth.The disclosed demand prediction operates over a range of time horizons.Short-term prediction on the order of minutes or hours can be useful forreal-time management of parking resources. On the other hand, long-termprediction on the order of days, weeks, or months is of value forlonger-term planning, such as hiring parking lot attendants, schedulingparking facility renovations, and so forth.

As disclosed herein, the goals of short-term versus medium-term versuslong-term prediction of off-street parking demand are difficult tosimultaneously satisfy using any one predictor function. Accordingly, insome embodiments demand for the off-street parking resource at aprediction time is estimated based on a model of historical occupancydata received from the off-street parking resource (which tends toprovide a baseline accurate in the longer term), and the demand for theoff-street parking resource at a prediction time is predicted byevaluating a regression function operating on an input data setincluding at least the demand at the prediction time estimated based onthe model and occupancy data received from the off-street parkingresource at k times prior to the prediction time. The adjustment byregression of depth k corrects for shorter-term deviations from thehistorical patterns captured by the model. The model may, for example,be generated using harmonic analysis generating a Fourier modelcomprises computing Fourier components for a plurality of differentperiods, such as a Fourier component with a period of one day, a Fouriercomponent with a period of one week, and/or a Fourier component with aperiod of one year. It is also contemplated to employ a Fouriertransform, e.g. implemented as a fast Fourier transform (FFT) or otherdiscrete Fourier transform (DFT), as the baseline model. A Fouriertransform approach generates the Fourier components as a function offrequency. The predictor function may, for example, comprise a supportvector regression (SVR) function of depth k trained on the history ofthe demand. Other predictor function embodiments are alternativelycontemplated, such as a predictor function embodied as a neural network,a regression tree, a k-nearest neighbors regressor, a Gaussian mixturemodel (GMM), or so forth, operating on an input data set including atleast the demand at the prediction time estimated based on the model andoccupancy data received from the off-street parking resource at k timesprior to the prediction time.

With reference to FIG. 1, an illustrative off-street parking resource 10comprises a multi-level parking garage. More generally, the off-streetparking resource may be a single- or multi-level parking garage, asurface parking lot, metered curbside parking, various combinationsthereof, or so forth. The off-street parking resource includes amechanism 12 for tracking occupancy of the off-street parking resource10. The occupancy may be measured directly. The illustrative trackingmechanism 12 is a ticket gate located at the entrance to the parkingfacility. In this approach, a driver approaching the entrance is stoppedby a lift-bar of the ticket gate, the driver obtains a parking ticketfrom the ticket gate, and this process causes the lift-bar toautomatically rise to admit the vehicle into the parking garage. It isassumed the vehicle parks, and therefore occupies the garage untilleaving. A similar arrangement can be provided at the parking garageexit, with vehicle exodus from the parking garage being detected by asuitable payment gate at the exit, operating automatically, or by a(human) gate attendant. The occupancy is then the time-aggregated valueof (vehicle admittances-minus-vehicle exits), with occupancy recorded ona chosen time basis, e.g. each hour, or in fifteen minute intervals, orso forth.

In some embodiments, recordation of vehicles entering the parkingfacility 10 via the entrance gate 12 is used as the occupancy metric,with the duration of stay in the facility being neglected. This approachis well-suited for parking resources that are closed at night, e.g. dayparking facilities or evening event parking facilities, where it may beexpected that most vehicles will stay until closing.

These are merely illustrative examples, and other approaches fortracking occupancy are contemplated, such as measuring time-in-operationof parking meters in the case of metered parking (e.g., occupancy equalsthe fraction of parking meters that do not have “time expired” status).

The output of the tracking mechanism 12 of the off-street parkingresource 10 is a time series of data. FIG. 1 illustrates the training ofthe demand prediction model, and accordingly the off-street parkingresource occupancy data define the occupancy data component 14 of a timeseries data training set 16, which may also include other relevant datasuch as illustrative weather data 18, or other relevant data such asgasoline prices, et cetera. The occupancy is suitably represented byvalues y. The goal of time series prediction is to estimate value y attime i based on past values at time i−1, i−2, . . . . For the regressionanalysis, the analysis may be limited in depth to the last k values(lag). Moreover, the time series prediction can also take into account anumber of external characteristics v_(i) at time i which may have impacton the series values, such as the illustrative weather data 18. Theobjective of the time series prediction is to find a functionŷ_(i)=ƒ(x_(i)), where x_(i)=(v_(i,1), . . . , v_(i,m), y_(i-k), . . . ,y_(i-1)) is an input data set, such that ŷ_(i), the predicted value ofthe time series at a future point in time, is consistent and minimize aregularized fit function. In the input data set x_(i), which may bewritten as a vector for mathematical convenience, the terms v_(i,1), . .. , v_(i,m) are relevant characteristics at time i, such as (in the caseof off-street parking) weather, gasoline price, and/or so forth.Moreover, the lag can also include some lag offset δ to take intoaccount that the most recent values may be unavailable at the time ofprediction. With a lag offset δ_(>)0, the input data may be written asx_(i)=(v_(i,1), . . . , v_(i,m), y_(i-k-δ), . . . , y_(i-1-δ).)

With continuing reference to FIG. 1, the occupancy data 14 of thetraining set 16 is input to a harmonic analysis module 20 to generatethe baseline model of the demand. In illustrative examples herein, theharmonic analysis module 20 performs Fourier series analysis todecompose the occupancy data 14 into a sum of a series of cosine andsine terms or, equivalently, into terms of the form A·cos(ωx+φ) where Ais an amplitude, ω is an angular frequency, and φ is a phase. Each termis defined by a unique amplitude A and a phase angle φ, where theamplitude value is half the height of the corresponding wave, and thephase angle (or simply, phase) defines the offset between the origin andthe peak of the wave over the range 0 to 2π. In one suitable formulationwith ƒ(x) being a continuous function on an interval [0, L], the Fourierseries representation for ƒ(x) can be written as:

$\begin{matrix}{{f(x)} = {{\sum\limits_{n = 1}^{\infty}\; \left( {{a_{n}\cos \frac{2\; \pi \; {nx}}{L}} + {b_{n}\sin \frac{2\; \pi \; {nx}}{L}}} \right)} + {\frac{1}{2}{a_{0}.}}}} & (1)\end{matrix}$

where in this formulation the a_(n) and b_(n) are amplitudes of thecosine and sine components, with the phase being related to these terms(e.g. if a_(n) is positive and b_(n)=0 then the phase is 0⁰; if b_(n) ispositive and a_(n)=0 then the phase is 90⁰; and so forth). By theorthogonality properties of sine and cosine, Equation (1) can bemanipulated to yield the following equations for a_(n) and b_(n), thecoefficients of the Fourier series:

$\begin{matrix}{{a_{n} = {{\frac{2}{L}{\int_{0}^{L}{{f(x)}\cos \frac{2\; \pi \; {nx}}{L}\ {x}\mspace{14mu} {for}\mspace{14mu} n}}} \geq 0}},} & (2) \\{b_{n} = {{\frac{2}{L}{\int_{0}^{L}{{f(x)}\sin \frac{2\; \pi \; {nx}}{L}\ {x}\mspace{14mu} {for}\mspace{14mu} n}}} \geq 0.}} & (3)\end{matrix}$

Additionally, b₀=0 is set, and a₀/2=1/L∫₀ ^(L)ƒ(x)dx, which is theaverage value of ƒ(x) over the interval [0, L]. Given a finite dataseries {y(k); k=1, . . . , N} instead of the continuous function ƒ(x)),a suitable approximation is the trapezoid approximation for a_(j) andb_(j), as follows:

$\begin{matrix}{{a_{j} = {{\frac{1}{N - 1}\left( {{y(1)} + {y(N)} + {2{\sum\limits_{k = 2}^{N - 1}\; {{y(k)}\cos \frac{2\; \pi \; {j\left( {k - 1} \right)}}{N - 1}}}}} \right)\mspace{14mu} {for}\mspace{14mu} j} \geq 0}},} & (4) \\{b_{j} = {{\frac{2}{N - 1}{\sum\limits_{k = 2}^{N - 1}\; {{y(k)}\sin \frac{2\; \pi \; {j\left( {k - 1} \right)}}{N - 1}\mspace{14mu} {for}\mspace{14mu} j}}} \geq 1.}} & (5)\end{matrix}$

In this discretized formulation, the frequency is given by the term 2πj.

The harmonic analysis module 20 generates a Fourier model 22, which isexpected to capture repetitive data cycles. In off-street parkingresource demand prediction, these cycles are typically expected tocorrespond to calendar associations 24 such as hourly cycles, dailycycles, weekly cycles, seasonal (e.g. yearly) cycles, or so forth. (Thisis also true of many other types of demand prediction problems, such asprediction of electrical power usage, retail sales, and so forth).Accordingly, the occupancy data 14 are tagged with the calendarassociations 24, and Equations (4) and (5) are evaluated for frequencyvalues corresponding to a plurality of different periods, typicallyincluding at least two of, and preferably all three of: one day (tocapture cycles over a 24 hour period); one week (to capture weeklycycles), and one year (to capture seasonal cycles). This entailsevaluating the coefficient terms a_(j), b_(j) for values for the term jcorresponding to these periods and integer multiples thereof. Theseasonal analysis is more effective when data for a complete year (ormore) is available and a similar sized data set is available as a testset.

With continuing reference to FIG. 1, a data vectors processing module 30constructs an input data set, suitably written as a vector, for eachoccupancy training datum of the time series training data set 16. Foroccupancy datum y_(i) at time i, the input data set is suitably of theform x_(i)=(y_(i) ^(H), v_(i-1), . . . , v_(i,m), y_(i-k-δ), . . . ,y_(i-1-δ)) where y_(i-k-δ), . . . , y_(i-1-δ) are the occupancy valuesof the occupancy data 14 previous to time i to depth k (with optionallag offset δ), v_(i,1), . . . , v_(i,m) are relevant characteristics attime i (e.g. weather in the illustrative example, more generally theterms may be omitted entirely in some embodiments), and y_(i) ^(H) isthe occupancy at time i as predicted by the Fourier model 22 or otherbaseline model generated by the harmonic analysis module 20. The set ofdata (x_(i), y_(i)), i=1, . . . , n form the training data input to apredictor function training module 32 that trains a predictor functionoperating on the input data set x_(i) including at least the estimateddemand y_(i) ^(H) for the resource at the prediction time i estimatedbased on the harmonic analysis and measured demand y_(i-k-n), . . . ,y_(i-1-n) for the resource measured at one or more times prior to theprediction time i. The predictor function may be a regression functionsuch as a support vector regression (SVR) function. The output of thistraining is a trained predictor function 34 for use in demandprediction.

With reference to FIG. 2, a demand prediction system 40, which has beentrained as described with reference to FIG. 1, receives time series data42 comprising occupancy data 44 and optional corresponding relevant datasuch as illustrative weather data 46. The demand prediction system 40operates to predict the demand y_(i) at a time i 50 for which occupancyvalues previous to time i to depth k with optional lag offset δ areincluded in the available data 40. A harmonic analysis prediction module52 applies the Fourier model 22 to generate a baseline demand predictiony_(i) ^(H). A vector construction module 54 then operates analogously tothe data vectors processing module 30 of the training phase (FIG. 1) togenerate an input data vector x_(i)=(y_(i) ^(H), v_(i,1), . . . ,v_(i,m), y_(i-k-δ), . . . , y_(i-1-δ)), where y_(i-k-δ), . . . ,y_(i-1-δ) are the occupancy values of the occupancy data 44 previous tothe time i 50 for which the prediction is to be made, to depth k withoptional lag offset δ. The values v_(i,1), . . . , v_(i,m), are relevantcharacteristics at prediction time i 50 (e.g. weather in theillustrative example), and baseline demand prediction y_(i) ^(H) outputby the harmonic analysis prediction module 52. The trained predictorfunction 34 generated by the training phase operates on this vectorx_(i) to generate the demand prediction 54.

In operation, the training phase (FIG. 1) and the inference orprediction phase (FIG. 2) are performed in an alternating fashion. Forexample, in one approach, an initial training phase performed asdescribed with reference to FIG. 1 is performed on historical data 16for parking occupancy of the off-street parking resource 10 to generatethe initial baseline model 22 and trained predictor function 34.Thereafter, this trained system is used to predict demand for theoff-street parking resource 10 as per the prediction system describedwith reference to FIG. 2. The system may in general be used to predictparking demand on a time horizon of hours, days, weeks, or longer. Aftersome time, e.g. once every week, the system reverts to the trainingphase to update the model (i.e. the baseline model 22 and the predictorfunction 34). In some embodiments, only training of the predictorfunction 34 is updated on a weekly (or other relatively frequent) basis,with the baseline model 22 being updated less frequently (e.g., on aper-year basis). Moreover, it is contemplated for updating of the modelto be performed based on accuracy feedback rather than (or in additionto) a regular, e.g. weekly, update schedule. For example, accuracy ofthe predicted demand ŷ_(i) can be determined when time i arrives and theactual demand y_(i) is measured. Thus, an error metric such as|ŷ_(i)−y_(i)| can be monitored, and when this error is too large (insome statistical sense over some sliding time window) then an update isperformed.

In system operation, the baseline model 22 captures the historicaltrends of daily, weekly, or seasonal cycles, while the trained predictorfunction 34 provides real-time adjustments for deviations fromhistorical trends. It is contemplated to train two or more predictorfunctions with different depths k and/or with different lag offsets δfor use in predictions over different time horizons. For example, apredictor may be trained with no lag offset (δ=0) and a small depth(e.g. k=1) to provide demand predictions on a daily basis (shorthorizon); whereas, demand predictions on a longer horizon, e.g. monthly,may employ a trained predictor function having a large lag offset δcorresponding to a month or a year, and a larger depth to providesmoothing. Advantageously, all such predictor functions optimized forthe different time horizons utilize the same baseline model 22 and thesame baseline demand prediction y_(i) ^(H) output by that model.

The systems of FIGS. 1 and 2 are suitably implemented by an illustrativecomputer 60 or other electronic data processing device. The inputtraining data 16 and time series data 42 are suitably provided by thetracking mechanism 12 automatically, or alternatively these data may beentered manually using a keyboard 62 or other user input device.Predicted demand ŷ_(i) may be displayed on a computer display 64 orother output device as numerical values, as a trend computed by runningthe demand prediction module 40 for each time 50 in an interval oftimes, or so forth. The systems of FIGS. 1 and 2 may additionally oralternatively be embodied by a non-transitory storage medium (not shown)storing instructions executable by the computer 60 or other electronicdata processing device to perform the disclosed training and predictionphases of the disclosed demand prediction. The non-transitory storagemedium may, for example, include one or more of the following: a harddisk or other magnetic storage medium; an optical disk or other opticalstorage medium; a random access memory (RAM), read-only memory (ROM),flash memory or other electronic storage medium; or so forth.

In the following, some illustrative examples of the predictor functionsuitably trained to generate the trained predictor function 34 aredescribed.

In one approach, the trained predictor function 34 comprises a supportvector regression (SVR) function trained on the history of the demand asfollows. Given the set of time series data 16 of the form (x_(i),y_(i)), i=1, . . . , n, the predictor function training module 32defines a function ƒ(x) that (when trained) will have an output close tothe predicted value for some prediction horizon. The prediction functionfor the linear regression is defined as ƒ(x)=w^(T)x+b. If the data isnot linear in its input space, the data x can be mapped into a higherdimension space, via a kernel function φ(x), and then the linearregression is performed in the higher dimensional feature space asƒ(x)=w^(T)φ(x)+b. The goal is therefore to find optimal weights w andthreshold b, as well as to define the criteria for finding an optimalset of weights.

Given training data (x_(i), y_(i)), i=1, . . . , n, x_(i)ε

^(d), y_(i)ε

, SVM first maps input vectors x onto the feature space Φ where φ(x)εΦand then approximates the regression by a linear functionƒ(x)=w^(T)φ(x)+b. This is obtained by solving the following optimizationproblem in the ε-insensitive tube (see, e.g. Smola et al., “A Tutorialon Support Vector Regression”, Statistics and Computing vol. 14 no. 3pages 199-222 (2004)):

$\begin{matrix}{{{{minimize}\mspace{14mu} \frac{1}{2}{w}^{2}} + {C{\sum\limits_{i = 1}^{n}\; \left( {\xi_{i} + \xi_{i}^{*}} \right)}}}{{subject}\mspace{14mu} {to}\left\{ \begin{matrix}{{y_{i} - \left( {{w^{T}{\varphi \left( x_{i} \right)}} - b} \right)} \leq {ɛ + \xi_{i}}} \\{{{\left( {{w^{T}{\varphi \left( x_{i} \right)}} + b} \right) - y_{i}} \leq {ɛ + \xi_{i}^{*}}},{i = 1},\ldots \mspace{14mu},n} \\{{\xi_{i}^{*} \geq 0},{\xi_{i} \geq 0},{i = 1},\ldots \mspace{14mu},n}\end{matrix} \right.}} & (6)\end{matrix}$

where ξ_(i), ξ_(i)*, i=1, . . . , n are slack variables, measuring thedeviation from ε-insensitive tube, and C is a regularization parameter.

In another approach, the trained predictor function 34 comprises anauto-regressive integrated moving average (ARIMA) model for time series.ARIMA is a generalization of an auto-regressive moving average (ARMA)model. The model is generally referred to as an ARIMA(p, d, q) modelwhere parameters p, d, and q are non-negative integers that refer to theorder of the auto-regressive, integrated, and moving average parts ofthe model respectively. An ARMA(p, q) model combines an auto-regressivemodel of order p with a moving average process of order q, whereas anARIMA(p, d, q) model also includes a differentiated component of orderd, to handle non-stationarity, that is:

φ(L)(1−L)^(d) y _(i)=ω(L)ε_(i),  (7)

where L is the back-shift (or lag) operator L^(k)X_(i)=X_(i-k), and φand ω are the two polynomials for the autoregressive (AR) and movingaverage (MA) components, and v_(t) is a noise.

To improve predictive accuracy, ARIMA models can be modified to takeinto account the periodic nature of time series data, an approach knownas a multiplicative seasonal ARIMA, or SARIMA, approach. It includesweekly or quarterly dependence relations within the auto-regressivemodel, by proving that the time series obtained as the differencebetween the observations in two subsequent weeks is weakly stationary.Conceptually, this approach is premised on the expectation that similarconditions typically hold at the same hour of the day and within thesame weekdays. The resulting SARIMA(p, d, q)×(P, D, Q)s model adds tothe standard ARIMA a seasonal auto-regressive, a seasonal movingaverage, and a seasonal differential component, as follows:

φ(L)Φ(L ^(S))(1−L)^(d)(1−L ^(S))^(D) y _(i)=ω(L)Ω(L ^(S))ε_(i),  (8)

where φ and ω are the two polynomials for the AR and MA components inEquation (8), Φ and Ω are the two polynomials for the seasonalcomponents.

The system for the demand prediction of FIGS. 1 and 2 combine a baselinehistory analysis (e.g. harmonic analysis generating a Fourier model)with a predictor function (e.g. SVR or another regression function, orARIMA or SARIMA) in order to provide more accurate prediction overvarious time horizons and time series data sets. In general, such acombination can be in a parallel mode in which outcomes of individualpredictors are combined through a weighted voting scheme, or (asillustrated in FIGS. 1 and 2) in a serial mode in which outputs of onepredictors (the harmonic predictor in illustrative FIGS. 1 and 2) areinput to the “downstream” predictor function 34.

In the following, some actually performed simulations are described. Inthese simulations, SVR is used as the last stage predictor (e.g. astrained predictor function 34), due to its capability to integratemultiple information sources. Predictions of auto-regressive andspectral (harmonic) methods are used as input observations for SVR(e.g., the output of the Fourier model 22). More generally, however, itis to be understood that the predictors in the serial combination may beneural networks, regression trees, k-nearest neighbors, Gaussianprocesses, or so forth.

The following the protocol was used in performing the experiments. Twostandard measures of errors were used, namely Mean Square Error (MSE)and Mean Average Percentage Error (MAPE), defined as:

$\begin{matrix}{{E_{MAPE} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}\; {\frac{y_{i} - {\hat{y}}_{i}}{y_{i}}}}}},{E_{MSE} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}\; \left( {y_{i} - {\hat{y}}_{i}} \right)^{2}}}},} & (9)\end{matrix}$

where y_(i) is the true value at time t in the test set while ŷ_(i) isthe prediction.

For machine learning techniques the time-series data were sequentiallysplit into training and test sets. However, this is not an idealsituation because of the different periodical trends present in thedata. For instance, if the training set spans over a regular period andthe test set incorporates a holiday season, this is likely to adverselyaffect the predictions since the test set follows a differentdistribution than the training set. In order to offset this problem andalso to analyze how the learning and testing handles the potentialdifferences, the percentage of data used as training set wasvaried—typically between 60% to 95% for each of the models. Performanceof these models is then compared based on these error curves.

Models based on auto-regressive techniques are typically useful forshort term predictions, especially if d>0, i.e. the model includesintegration term I(d). The prediction error quickly accumulates sincethe predictions are made based on a fixed number of previousobservations. For long-term predictions, training a singleauto-regressive model may give poor performance because of theindependence of these models from test sets. Parameters learned duringthe learning phase are not applied to a test set; but are applied,sequentially, to the training set and finally to new predictions made.For this reason, for auto-regressive models, several models wereincrementally learned and predictions were made on small intervals.These predictions were then combined and performance measures calculatedfrom these predictions based on several models instead of a singlemodel.

Experiments were performed on off-street parking resource occupancy datagathered for parking facilities at the Toulouse Blagnac Airport inFrance (hereinafter “Blagnac”). The primary tracking mechanism 12 forcollecting the occupancy data in the form of transactional data washardware equipment which record different events such as entry and exitat Blagnac airport parkings. Transactional records on each vehicle entryand exit were transformed and aggregated into time sequences at aspecific (fixed) duration time. Sequences were generated at 15 minutes,30 minutes intervals as well as 1 hour intervals.

With reference to FIG. 2, two datasets were collected, covering periodsfrom February 2012 to mid-May 2012, and from December 2012 to mid-March2013. Each data set comprises data from five airport parking spaces.Parking P0 is considered of no interest since it is the drop lane at theairport and stationing of vehicles beyond 30 minutes is prohibited. Ofthe remaining four parking facilities, two pairs, each with similarpricing and location, were of primary interest for the analyses reportedherein. Off-street parking resources P1 and P2 are designated asshort-term parking facilities which are relatively closer to the airportterminals but are relatively more expensive. Off-street parkingresources P5 and P6 are relatively further away from the terminals, butare also relatively cheaper and are designated as long-term parking. Thecollected hourly occupancy data for both types of parking and for bothdatasets are plotted in FIG. 3.

As previously noted, the occupancy data may be augmented by otherrelevant data, such as weather data in the illustrative examples ofFIGS. 1 and 2. Weather conditions may have an impact on the parkingoccupancy, e.g. in poor weather it may be expected that closer (buthigher priced) parking facilities may be preferred over more distant(albeit cheaper) parking facilities. Also, some of the parkings whichare farther from the airport are open air, and it is necessary to take abus to arrive at the terminals. Accordingly, the Blagnac dataset isaugmented by collected historic hourly weather data corresponding to theoccupancy data. Several parameters, such as temperature, wind-speed,windchill, precipitation and humidity are considered. These datacorrespond to some terms v_(i,1), . . . , v_(i,m), of input data setx_(i)=(y_(i) ^(H), v_(i,1), . . . , V_(i,m), y_(i-k-δ), . . . ,y_(i-1-δ).)

Calendar information 24 corresponding to the occupancy data are alsocollected, such as the day of the week and whether there are holidaysduring a particular period. This is useful as the patterns followed byusers of the parking facilities are expected to be are different onweekdays and weekends and during vacation periods. Information regardingthe hour of day is also included as an input observation, i.e. as someof the terms v_(i,1), . . . , v_(i,m).

In time series analysis, it is often the case that observing the lastobservations gives often a good idea for prediction the next value. If,for example, the occupancy during the last three hours has beenincreasing, it is likely to continue the trend. The occupancy at anygiven time is expected to be highly correlated with the most recentoccupancy observations. The number of lags k which are helpful inpredicting the occupancy may vary for different parking spaces due tothe inherent differences in occupancy patterns of these parking spaces.In most experiments reported herein, the prediction models were builtwith lag k=3 or k=5. Additionally, a lag offset δ may be included so asto go beyond the most recent observations. Lag offsets of 24 hour, 48hour and 168 hour were tested. This facilitates capture of cyclicbehavior of occupancy. Time lags of 24 and 48 hours model theexpectation that occupancy of parkings at the same hour of the day issimilar across days. While this is likely to be true for some days, itmight not be true in all cases; for instance, occupancy values arewidely different between weekdays and weekends. In order to eradicatethis, lag offsets of 168 hour were also tested, which corresponds to theparking occupancy one week ago on the same day of week.

Based on the foregoing approach, different prediction scenarios weretested. Models were trained for use in immediate prediction, i.e.assuming that at time i, the occupancy observations at time i−1, i−2, .. . are available, as well as all external characteristics. In otherwords, these immediate prediction models assume lag offset δ=0. Inmid-term prediction scenarios, no immediate occupancy observation isavailable, and so some lag offset δ>0 is used. This reflects practicalcases in which real-time data registered at various physical locationsand devices take minutes, hours and even days to be collected andprocessed. Finally, in long-term prediction cases, the decision makersexpect a demand estimation weeks and months in advance.

With reference to FIG. 4, the harmonic analysis module 20 processed theBlagnac data as follows. The Fourier components for a period of one daywere first computed for the occupancy data 14. The resulting model wassubtracted from the occupancy data 14 to generate residual data. TheFourier components for week and season (e.g. year) intervals were thencomputed for this residual data. The resulting harmonic analysis of thefirst Blagnac dataset is shown in FIG. 4, which plots the true occupancydata and the daily, weekly and seasonal cycles, along with how the sumof three cycles fits the read data, and the residual data.

With reference to FIG. 5, in one set of experiments Moving Average (MA)models were considered. FIG. 5 shows the errors obtained for differentlag values, using the auto-regressive protocol. It is observed that bestresults are achieved for the lag k=1. As can be seen from FIG. 5, themodel with just one lag performs well with MAPE 0.97% for short term(ST) and 3.84% for long term (LT) parkings.

In further experiments, ARIMA models were learned using theauto-regressive protocol. Models were incrementally trained and used tomake predictions over a short period of time. In this way, models arecomparable to regression models whose results are also reported here.However, the ARIMA predictions are only reasonable for short periods oftime (1-2 hours); for longer time periods, the ARIMA predictionsdeteriorate with large confidence intervals. Thus, for the Blagnac dataset ARIMA is suitable for short-term prediction but not for longer term(e.g. greater than 1-2 hour) prediction horizons.

With reference to FIGS. 6 and 7, tests using SVR as the predictorfunction are reported. FIGS. 6 and 7 report MSE and MAPE values for SVRmodels trained with 3 lags, 5 lags, 24 Hr, 48 Hr and 168 Hrobservations, as well as day-of-week and Harmonics features. Each curveplots the errors against the percentage of the data used as the testset, for both short-term (ST) and long-term (LT) parkings. Modelstrained with immediate occupancy observations yielded the most accuratepredictions. Using lag k=5 performs better for LT parkings, while lagk=3 are sufficient for the case of ST parkings. MAPE values show thatoccupancy feedback is a suitable predictor with average prediction errorjust above 1% for ST parkings while it is marginally below 1% in case ofLS parkings with lag k=5. Occupancy values from the past week (168Hours), at the same time of the day is also a good predictor for STparkings with low MSE and MAPE between 2.5-3.0% for all training setsizes. Observations of the day of the week (DOW) comprise seven mutuallyexclusive binary indicator variables representing the day of the week.The SVR also performs well for the DOW data, with similar error rates asthe one week delay observations. For LT parkings, 24 hour and 48 hourlagged observations are better predictors than the Day-of-Week and1-Week lagged observations. This suggests that ST parkings of theBlagnac data sets have a more regular pattern based on the day of weekand daily trend which is independent of the occupancy status of theparking from previous days. By contrast, the LT parking data exhibits amore relative trend based on the occupancy over the past few days. Thismay reflect that ST parkings are mostly used for short parking stays,while LT parkings are used for longer trips.

With reference to FIGS. 8 and 9, in another experiment, multiplecharacteristics and observations were combined to make an immediateprediction. Results for some of these combinations are plotted in FIGS.8 and 9. As seen in these plots, extending the lag k=3 model withHour-of-Day feature substantially reduces the MAPE for ST parkings. Forthe LT parkings, using the harmonics analysis or 48 Hr laggedobservation reduces the error values, particularly when we dispose lesstraining data (the training data percentage is inferior to 75%).

With reference to FIGS. 10 and 11, demand prediction on a mid-termhorizon was tested for the Blagnac data sets. This scenario assumes ashort delay in availability of occupancy data. In experiments, thelatest data was assumed to be available with 1 or 2 hours of delay. Dueto the delay, all models behave slightly worse in terms of predictionquality compared to the immediate case. FIG. 10 plots MSE and MAPEvalues for ST parkings, when the delay is one hour (corresponding to lagoffset δ=1, that is, prediction for y_(i) can observe values y_(i-2),y_(i-3), . . . ). The lag k=3 model performs well, and an improvement isobtained by combining k=3 it with the Hour-of-Day feature. FIG. 11 plotserror values for LT parkings. It is seen that the delayed lags k=3perform well, with further improvements achieved when combined with1-Week lagged observation or with harmonic analysis.

With reference to FIG. 12, demand prediction on a long-term horizon wastested for the Blagnac data sets. In this scenario, the currentobservations and features are used to make one week forecast. As FIG. 12reports, the occupancy is predicted on this time horizon with about2.5%-3.0% MAPE which can further be improved with the use of HarmonicAnalysis to reach under 2% for most training set sizes.

Demand prediction techniques disclosed herein employ differentcombinations of base predictors. In the experiments on the Blagnac datasets, serial composition of harmonic analysis with SVR was found toperform well, including in long prediction scenarios. Othercombinations, such as the weighted voting or a serial composition ofARIMA models and SVR, are also contemplated. Table 1 compares MAPEvalues for some of the previously tested models on Blagnac dataset, withand without input of ARIMA. No positive impact of using ARIMA models inthe serial mode is observed.

TABLE 1 Model MAPE 3 Lags 0.010372 3 Lags; with ARIMA 0.012218 HA2; 24Hr; 48 Hr 0.020108 HA2; 24 Hr; 48 Hr; with ARIMA 0.016881

With reference to FIG. 13, another experiment was performed, which useda weighted voting schema on predictions of previously discussed models.It used a linear combination of predictions from ARIMA and SVR models,which can be represented formally as ŷ_(hybrid)=αyarima+(1−α)y_(svr).However, in this case, SVR model always performs better and thecombination did not exhibit any advantages over the SVR model. FIG. 13plots the MAPE when the weight for ARIMA model is varied between 0.01and 0.99.

The evaluation tests on Blagnac off-street parking dataset demonstratedefficacy of the demand prediction methods disclosed herein for variouspractical scenarios. In the case of immediate prediction, ARIMA and SVRboth perform well, with SVR being more flexible in integrating multiplesources and requiring less re-training than ARIMA. For long-termprediction, spectral models effectively capture multiple cyclic trendsin data. In mid-term scenarios, SVR is robust against data collectiondelay (i.e. lag offset δ), and facilitates integrating predictions ofother models as the downstream predictor function of a serialcombination.

It will be appreciated that various of the above-disclosed and otherfeatures and functions, or alternatives thereof, may be desirablycombined into many other different systems or applications. Also thatvarious presently unforeseen or unanticipated alternatives,modifications, variations or improvements therein may be subsequentlymade by those skilled in the art which are also intended to beencompassed by the following claims.

1. A non-transitory storage medium storing instructions readable andexecutable by an electronic data processing device to perform a methodcomprising: estimating demand for a resource at a prediction time basedon a harmonic analysis of a history of the demand; and generating aprediction of demand for the resource at the prediction time byevaluating a predictor function operating on an input data set includingat least the estimated demand for the resource at the prediction timeestimated based on the harmonic analysis and measured demand for theresource measured at one or more times prior to the prediction time. 2.The non-transitory storage medium of claim 1 wherein the resource is oneof: off-street parking, electrical power, gasoline, and a retailmerchandise product.
 3. The non-transitory storage medium of claim 1wherein the resource is off-street parking and the input data setfurther includes weather data measured at the one or more times prior tothe prediction time.
 4. The non-transitory storage medium of claim 1wherein the predictor function comprises a support vector regression(SVR) function trained on the history of the demand.
 5. Thenon-transitory storage medium of claim 1 wherein the predictor functioncomprises a regression function trained on the history of the demand andthe method further comprises: generating a Fourier model of the historyof the demand, wherein the estimating comprises estimating demand forthe resource at the prediction time by evaluating the Fourier model atthe prediction time; and training the regression function on the historyof the demand.
 6. The non-transitory storage medium of claim 5 whereinthe generating of the Fourier model comprises computing Fouriercomponents for a plurality of different periods including at least twoof (1) one day, (2) one week, and (3) one year.
 7. The non-transitorystorage medium of claim 5 wherein the generating of the Fourier modelcomprises computing a Fourier transform of the history of the demand. 8.An apparatus comprising: an electronic data processing device configuredto receive occupancy data from an off-street parking resource, theelectronic data processing device further configured to: estimate demandfor the off-street parking resource at a prediction time based on amodel of historical occupancy data received from the off-street parkingresource; and generate a predicted demand for the off-street parkingresource at a prediction time by evaluating a regression functionoperating on an input data set including at least the estimated demandfor the off-street parking resource at the prediction time estimatedbased on the model and occupancy data received from the off-streetparking resource at k times prior to the prediction time.
 9. Theapparatus of claim 8 wherein the input data set further includes weatherdata measured at the k times prior to the prediction time.
 10. Theapparatus of claim 8 wherein the regression function comprises a supportvector regression (SVR) function trained on the historical occupancydata.
 11. The apparatus of claim 8 wherein the electronic dataprocessing device is further configured to generate said model as aFourier model of the historical occupancy data.
 12. The apparatus ofclaim 11 wherein the electronic data processing device is configured togenerate the Fourier model of the historical occupancy data byoperations including computing a Fourier component for a time intervalof one day and computing a Fourier component for a time interval of oneweek.
 13. The apparatus of claim 11 wherein the electronic dataprocessing device is configured to generate the Fourier model of thehistorical occupancy data by computing a Fourier transform of thehistorical occupancy data.
 14. The apparatus of claim 8 wherein theelectronic data processing device is further configured to train theregression function on the historical occupancy data.
 15. A methodcomprising: modeling a history of demand for a resource to generate abaseline model of the demand; predicting demand for the resource at aprediction time by evaluating a regression function of depth k operatingon an input data set including at least the demand for the resource atthe prediction time output by the baseline model of the demand andmeasured demand for the resource measured at k times prior to theprediction time, where k is a positive integer greater than or equal toone; wherein the modeling and the predicting are performed by anelectronic data processing device.
 16. The method of claim 15 wherein kis greater than or equal to two.
 17. The method of claim 15 wherein theresource is off-street parking and the input data set further includesweather data measured at the one or more times prior to the predictiontime.
 18. The method of claim 15 further comprising: training theregression function on the history of demand for the resource, whereinthe training is performed by the electronic data processing device andutilizes the baseline model of the demand.
 19. The method of claim 18wherein the regression function comprises a support vector regression(SVR) function.
 20. The method of claim 15 wherein the baseline modelcomprises a Fourier model and modeling comprises: computing Fouriercomponents of the history of demand for the resource.